Question

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Answer 1

Question :-

$\rm{ \displaystyle \lim_{x \to 0 } \rm{\frac{ \sqrt{a + x} - \sqrt{a} }{x} }}$

Solution :-

Given :

$\rm{ \displaystyle \lim_{x \to 0 } \rm{\frac{ \sqrt{a + x} - \sqrt{a} }{x} }}$

$\rm{as \: x \to 0 \: it \: is \: \frac{0}{0} \: \: form }$

So using Rationalization,

$\sf:\longmapsto{\displaystyle \lim_{x \to 0} \rm{\frac{ \sqrt{a + x} - \sqrt{a} }{x} } \times \frac{ \sqrt{a + x} + \sqrt{a} }{ \sqrt{a + x} + \sqrt{a} } }$

$\sf:\longmapsto{\displaystyle \lim_{x \to 0} \rm{ \frac{ (\sqrt{a + x} - \sqrt{a} )( \sqrt{a + x} + \sqrt{a}) }{x( \sqrt{a + x} + \sqrt{a} )} }}$

$\sf:\longmapsto{\displaystyle \lim_{x \to 0} \rm{ \frac{( \sqrt{a + x } ) {}^{2} - ( \sqrt{a} ) {}^{2} }{ \sqrt{ax + x {}^{2} } + \sqrt{ax} } }}$

$:\longmapsto{\displaystyle \lim_{x \to 0} \rm{ \frac{a + x - a}{2 \sqrt{ax } + x} } }$

As we know x = 0/0 = 1

$:\longmapsto{\displaystyle \lim_{x \to 0} \rm{ \frac{1}{2 \sqrt{ax} + x} }}$

Hence,

$\bigstar \: {\boxed{\rm{\red{\displaystyle \lim_{x \to 0} \rm{ \frac{1}{2 \sqrt{ax} + x} }}}}}$

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Answer 2

Answer:

Step-by-step explanation:

1/2√a

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